If the side of an equilateral triangle is 'a', then area of this triangle is equal to:

Options

A. `(3a^2)/2`
B. `sqrt(3a^2)/2`
C. `sqrt(3a^2)/4`
D. `sqrt(3a^2)`

Answer

To find the area of an equilateral triangle with side length 'a', we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Triangle**:
- An equilateral triangle has all three sides equal and all angles equal to 60 degrees.

2. **Using the Area Formula**:
- The area \( A \) of any triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
- Alternatively, for two sides and the included angle, the area can be calculated as:
\[ A = \frac{1}{2} \times S_1 \times S_2 \times \sin(\theta) \]
- Here, \( S_1 \) and \( S_2 \) are the lengths of two sides, and \( \theta \) is the angle between them.

3. **Applying the Formula to the Equilateral Triangle**:
- In our case, both sides \( S_1 \) and \( S_2 \) are equal to \( a \), and the angle \( \theta \) is 60 degrees.
- Therefore, we can substitute these values into the formula:
\[ A = \frac{1}{2} \times a \times a \times \sin(60^\circ) \]

4. **Calculating \( \sin(60^\circ) \)**:
- The value of \( \sin(60^\circ) \) is \( \frac{\sqrt{3}}{2} \).
- Substituting this value into the area formula gives:
\[ A = \frac{1}{2} \times a^2 \times \frac{\sqrt{3}}{2} \]

5. **Simplifying the Expression**:
- This simplifies to:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

### Final Result:
- The area of the equilateral triangle with side length \( a \) is:
\[ A = \frac{\sqrt{3}}{4} a^2 \]